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- $x^a - y^b = 1$
for $a, b > 1$ and $x, y > 0$, is:
- $x = 3, a = 2, y = 2, b = 3$
Also known as
- Consecutive Integers which are Powers of 2 or 3: the special case where $x$ and $y$ are $2$ and $3$
- 1 plus Power of 2 is not Perfect Power except 9: the special case of $y = 2$.
- 1 plus Perfect Power is not Power of 2: the special case of $x = 2$.
- 1 plus Square is not Perfect Power: the special case of $b = 2$.
- 1 plus Perfect Power is not Prime Power except for 9: the special case where $x$ is prime.
Source of Name
This entry was named for Eugène Charles Catalan.
It was proven in $2002$ by Preda V. Mihăilescu.
- 1944: Eugène Catalan: Note extraite d'une lettre adressée à l'éditeur (J. reine angew. Math. Vol. 27: p. 192)
- 2004: Preda Mihăilescu: Primary Cyclotomic Units and a Proof of Catalan's Conjecture (J. reine angew. Math. Vol. 572: pp. 167 – 195)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Catalan's conjecture